Conditions for absence of global SU(2) anomaly for four-dimensional gauge groups

Proofs are given group theoretically for the absence of the global SU(2) (Witten) anomaly for any compact Lie group G⊃SU(2) satisfying ${\Pi }_4$(G)=0 (${\Pi }_4$ is the four-dimensional homotopy group). It is shown that the number of fermion zero modes is even for G=SO(N) (N≥7) and the five exceptional groups ${\mathrm{G}}_2$, ${\mathrm{F}}_4$, ${\mathrm{E}}_6$, ${\mathrm{E}}_7$, and ${\mathrm{E}}_8$. The absence of the Witten anomaly for the SU(N) (N≥3) groups requires freedom from the perturbative (triangle) anomaly.